Everybody buy now should know, that one should pull crypto assets off exchanges when done with trading. It’s quite easy: Put them back in your wallet off the exchange (preferably an off-line wallet) and you are done. What about fiat currency on exchanges? It’s a) expensive and b) time consuming to pull them off after selling crypto, if one intends to continue trading. On the other hand, there isn’t more reason to trust the exchange with your $, €, etc. Any suggestions?

I’d say that generally it is safer to leave crypto on an exchange. There are two main reasons for this:

1) There are decades, if not centuries, of laws governing the custodianship of fiat money. As much as exchanges make our lives miserable with AML & KYC requirements, I’m sure they’re subject to similarly burdensome rules governing how they manage and store client funds.
If an exchange refuses to return your fiat funds, they can be sued for it and the funds returned by the banks – this is not true of crypto unless the exchange’s private keys are seized.

2) If stolen, fiat money can in most cases be frozen or clawed back via the banking system. If you report your credit card as stolen, it’s easy for the bank to freeze the account. The same is true for exchange accounts.

Further, many exchanges have insurance for their clients’ fiat. funds I would say it’s a lot harder to get insurance for client crypto. It seems crypto would be hard to insure because its price is so volatile as to require frequent re-evaluation, and also because insurers are unlikely to want to deal with it as (to them) it’s such a complex and poorly-understood asset.